Let 1 ≤ p ≤ q and suppose that for 1 ≤ j ≤ n we have xj ≥ 0. Prove that
( n ∑ j=1 (xj^q) )^ (1/q) ≤ ( n ∑ j=1 (xj^p) )^ (1/p) ≤ n^ (1/p - 1/q) ( n ∑ j=1 (xj^q) )^ (1/q)
Let 1 ≤ p ≤ q and suppose that for 1 ≤ j ≤ n we have xj ≥ 0. Prove that ( n ∑ j=1 (xj^q) )^ (1/q)...
real analysis proof Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1 Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1
Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E. Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E.
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1 +o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
2.Let Xj,X,, Xj, X4, Xj be a random sample of size n-5 from a Poisson distribution with mean ?. Consider the test Ho : ?-2.6 vs. H 1 : ? < 2.6. a)Find the best rejection region with the significance level a closest to 0.10 b) Find the power of the test from part (a) at ?= 2.0 and at ?=1.4. c) Suppose x1-1, x2-2, x3 -0, x4-1, x5-2. Find the p-value of the test.
Suppose that the covariates Xj,i for i 1, 2, , n and j 1, 2, , indicator variables for a single categorical variable in the manner covered in the course. Thus, suppose that for each individual i = 1,2,…,n we have that X1.i, X2.i,...,Xd,i this one is equal to the number 1. Let Bk be the (A , . . . , β 1), the minimizer of L (bi , b2, . . . ,勿of eq. (B. = Yn.(k), where...
F1 If, in the Fibonacci search algorithm. when ,f(p) /(q) we let p = b-- (instead of b-2c), prove that p = q Likewise, if when f(p) > f(q) we let q - a+(a) (instead of a +2e), prove that p q. | (b-a) F1 F2 F1 If, in the Fibonacci search algorithm. when ,f(p) /(q) we let p = b-- (instead of b-2c), prove that p = q Likewise, if when f(p) > f(q) we let q - a+(a)...
Let P be some probability measure on sample space S = [0, 1]. (a) Prove that we must have limn→∞ P((0, 1/n) = 0. (b) Show by example that we might have limn→∞ P ([0, 1/n)) > 0.
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of heads after n-tosses so (X·= k)-(there are k heads after n tosses of the coin). (a) Compute the P(Y> n) (b) Prove using the formula P(AnB) P(B) (c) What is the physical meaning of the formula you just proved? Suppose...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
Covariance: Suppose 10 balls are randomly distributed into 4 urns. Let Xj be the number of balls that fall into the jth urn, and let Ti be the indicator variable that the ith ball falls into jth urn, with E {1, . . . , 10} and j є {1, . . . , 4). ·P(X, = 0) =? (hird